Sources and References

Primary textbook — Svelto, O. (2010). Principles of Lasers, 5th ed. Springer. (Available in UW library.) Supplementary texts — Saleh, B. E. A., & Teich, M. C. (2019). Fundamentals of Photonics, 3rd ed. Wiley; Siegman, A. E. (1986). Lasers. University Science Books; Yariv, A. (1989). Quantum Electronics, 3rd ed. Wiley. Online resources — MIT OpenCourseWare 6.007 Applied Electromagnetics; Strickland & Mourou (1985) CPA paper (Opt. Commun. 56, 219–221) for historical context on chirped pulse amplification; NIST Atomic Spectra Database (physics.nist.gov/asd).

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Chapter 1: Overview — Lasers, Light, and the Electromagnetic Spectrum

1.1 Why Study Light-Matter Interactions?

The interaction of electromagnetic radiation with matter underlies an enormous range of technologies: lasers, optical fibers, LEDs, solar cells, spectroscopy, medical imaging, and nonlinear optics. The central question is: when electromagnetic radiation is incident on a material, how does the material respond, and how does that response modify the radiation?

Before the laser (1960), light sources were thermal (incoherent, broadband, low intensity). The laser provided coherent, highly monochromatic, well-collimated, high-intensity radiation that enabled nonlinear optics, precision measurement, and ultrafast science.

1.2 Electromagnetic Waves and Photons

Classically, light is a transverse electromagnetic wave: oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation. The dispersion relation in vacuum is \( \omega = ck \), where \( c = 3 \times 10^8 \, \text{m/s} \).

\[ E = \hbar \omega = h\nu \]

and momentum \( p = \hbar k \). For optical frequencies (\( \lambda \sim 500 \, \text{nm} \)), \( E \approx 2.5 \, \text{eV} \).

1.3 The Spectrum

Region	Wavelength range	Photon energy
X-ray	0.01–10 nm	keV range
UV	10–400 nm	3–100 eV
Visible	400–700 nm	1.8–3.1 eV
Near-IR	0.7–2.5 μm	0.5–1.8 eV
Mid-IR	2.5–25 μm	0.05–0.5 eV
Far-IR / THz	25 μm–1 mm	meV range

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Chapter 2: Semi-Classical Theory — Absorption, Stimulated, and Spontaneous Emission

2.1 Atoms in a Radiation Field

The interaction of an atom with a near-resonant electromagnetic field is treated semi-classically: the atom is described quantum mechanically, while the field is treated classically (a sinusoidal electric field). This yields the foundational Einstein relations.

Consider a two-level atom with ground state \( \left| 1 \right\rangle \) (energy \( E_1 \)) and excited state \( \left| 2 \right\rangle \) (energy \( E_2 \)), with the resonance frequency \( \omega_0 = (E_2 - E_1)/\hbar \).

2.2 Einstein Coefficients

\[ \frac{A_{21}}{B_{21}} = \frac{\hbar \omega^3}{\pi^2 c^3} \]\[ B_{12} = B_{21} \quad (\text{for non-degenerate levels}) \]

The ratio \( A_{21}/B_{21} \propto \omega^3 \) explains why spontaneous emission dominates over stimulated emission at high frequencies (X-rays are hard to amplify) and why lasers work best in the optical/IR range.

2.3 Transition Rate Probability (Fermi’s Golden Rule)

\[ W_{12} = \frac{2\pi}{\hbar} \left| \langle 2 | \hat{H}' | 1 \rangle \right|^2 \rho(E_2) \]

For electric dipole transitions, the matrix element is \( \langle 2 | \hat{H}' | 1 \rangle = -e \mathbf{E} \cdot \langle 2 | \mathbf{r} | 1 \rangle \). Forbidden if the matrix element vanishes (selection rules).

The dipole matrix element \( \mu_{12} = e \langle 2 | \mathbf{r} | 1 \rangle \) determines the strength of the transition. Large \( |\mu_{12}|^2 \) means large \( B_{12} \) and short spontaneous lifetime \( \tau_{sp} = 1/A_{21} \).

2.4 Rate Equations for Population Densities

Let \( N_1 \), \( N_2 \) be the population densities (atoms/m³) in states 1 and 2. With total density \( N = N_1 + N_2 \):

\[ \frac{dN_2}{dt} = B_{12} \rho N_1 - B_{21} \rho N_2 - A_{21} N_2 \]\[ \frac{dN_1}{dt} = -\frac{dN_2}{dt} \]\[ \Delta N = -N \frac{A_{21}}{A_{21} + 2B_{21}\rho} \]

For a two-level system, \( \Delta N < 0 \) always: stimulated emission can at most equalize populations but cannot achieve population inversion (\( N_2 > N_1 \)). This is why two-level lasers are not feasible — we need three or four levels.

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Chapter 3: Laser Gain and Oscillation

3.1 Population Inversion and Gain

\[ g(\omega) = \sigma(\omega) \Delta N \]

where \( \sigma(\omega) \) is the stimulated emission cross-section (m²) and \( \Delta N = N_2 - N_1 \). When \( \Delta N > 0 \), \( g > 0 \): the medium amplifies the field.

\[ \sigma(\omega) = \frac{\hbar \omega}{c} B_{21} g(\omega) \]

where \( g(\omega) \) here is the normalized lineshape function (not gain).

3.2 Three-Level Laser System

Example: Ruby (\( \text{Cr}^{3+}:\text{Al}_2\text{O}_3 \)) — Maiman’s first laser (1960).

1. Pumping: atoms are pumped from level 1 (ground) to a broad upper pump band (level 3) by flash lamp.
2. Rapid non-radiative decay: level 3 → level 2 (upper laser level) via phonon emission (fast).
3. Lasing transition: level 2 → level 1 (lower laser level = ground state) with emission at the laser wavelength.

Problem: the lower laser level is the ground state, so more than half the atoms must be pumped out of the ground state to achieve inversion. High pump threshold.

\[ \frac{dN_2}{dt} = R_p N_1 - \frac{N_2}{\tau_{21}} - B \rho \Delta N \]

Threshold condition: \( R_p > 1/(2\tau_{21}) \) for inversion.

3.3 Four-Level Laser System

Example: Nd:YAG, He-Ne, CO₂.

1. Ground state (level 0) → pump band (level 3): fast pumping.
2. Level 3 → upper laser level (level 2): fast non-radiative decay.
3. Lasing transition: level 2 → lower laser level (level 1).
4. Level 1 → ground state (level 0): fast non-radiative decay, keeping level 1 nearly empty.

Advantage: lower laser level is essentially empty → inversion achieved with much lower pump power than three-level system. Most practical lasers use four-level or quasi-four-level schemes.

3.4 Saturation of the Gain and Laser Oscillation

\[ g(\phi) = \frac{g_0}{1 + \phi/\phi_s} \]

where \( g_0 \) is the small-signal gain and \( \phi_s \) is the saturation photon flux.

\[ G_{rt} = \exp\!\left[ 2 g_0 L \right] \geq \frac{1}{(1-T)(1-\alpha_i)^2} \]\[ g_0 \geq \frac{\delta_{total}}{2L} \]

where \( \delta_{total} \) captures all round-trip loss (output coupling, internal scattering, absorption, diffraction).

Above threshold, the photon flux builds up until gain saturation clamps \( g(\phi) = \delta_{total}/2L \). This is the steady-state operating point.

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Chapter 4: Coherence

4.1 What is Coherence?

Coherence determines whether a light source can produce stable interference fringes and is the key property distinguishing laser light from thermal (incoherent) sources.

4.2 Temporal Coherence

CW lasers: single-mode He-Ne laser has \( \Delta \nu \sim 1 \, \text{kHz} \), giving \( \ell_c \sim 300 \, \text{km} \). Thermal source (sodium lamp): \( \Delta \nu \sim 1 \, \text{GHz} \), \( \ell_c \sim 30 \, \text{cm} \).

\[ \Delta \nu \cdot \Delta t \geq \frac{K}{2\pi} \]

where \( K \sim 0.44 \) (Gaussian pulse). A 1 ps transform-limited pulse has \( \Delta \nu \sim 440 \, \text{GHz} \).

The time-bandwidth product is a Fourier relationship: shorter pulses require broader spectra. This is not an uncertainty principle but a classical wave property.

4.3 Spatial Coherence

A spatially coherent source can be collimated into a parallel beam (small divergence). Lasers in the fundamental TEM₀₀ mode have essentially perfect spatial coherence.

\[ \theta_d = \frac{\lambda}{\pi w_0} \quad \text{(half-angle)} \]

Thermal sources (incandescent bulbs) have negligible spatial coherence: light is emitted from many independent emitters; diverges in all directions; cannot be focused to a diffraction-limited spot.

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Chapter 5: Laser Cavities — Stability and Modes

5.1 The Optical Resonator

A laser cavity (resonator) consists of two or more mirrors that trap light and provide feedback for amplification. Most common: Fabry-Pérot cavity with two mirrors separated by length \( L \).

Functions:

1. Provides optical feedback (recirculates photons through the gain medium).
2. Selects cavity modes (specific frequencies and spatial profiles are supported).
3. Determines the output beam properties.

5.2 Cavity Stability

\[ -1 \leq \frac{A+D}{2} \leq 1 \]\[ g_1 = 1 - \frac{L}{R_1}, \quad g_2 = 1 - \frac{L}{R_2} \]

Stability condition: \( 0 \leq g_1 g_2 \leq 1 \).

Special cases:

Configuration	\( g_1, g_2 \)	Stability
Plane–plane	1, 1	Marginally stable
Concentric	−1, −1	Marginally stable
Confocal	\( g_1 = g_2 = 0 \)	Marginally stable (but robust to misalignment)
Hemispherical	\( g_1 = 1, g_2 = 0 \)	Marginally stable
General curved	Various	Stable if \( 0 < g_1 g_2 < 1 \)

5.3 Cavity Modes

\[ \nu_q = \frac{qc}{2L}, \quad q = 1, 2, 3, \ldots \]

Mode spacing: \( \Delta\nu_{FSR} = c/(2L) \) (free spectral range). For \( L = 30 \, \text{cm} \), \( \Delta\nu_{FSR} = 500 \, \text{MHz} \).

The laser will oscillate at longitudinal modes whose frequency falls within the gain bandwidth of the medium. Single-mode operation (one longitudinal mode) is achieved by short cavities or intracavity etalons.

Transverse (spatial) modes — TEM₍ₘₙ₎: Spatial intensity patterns across the beam cross-section, labelled by indices \( m, n \) (Hermite-Gaussian) or \( p, l \) (Laguerre-Gaussian). The TEM₀₀ mode is Gaussian (no nodes) and has the highest beam quality (\( M^2 = 1 \)).

Higher-order modes have larger cross-sections and more complex spatial structure; apertures inside the cavity preferentially suppress higher-order modes.

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Chapter 6: Confocal Cavity

6.1 Confocal Resonator Analysis

The confocal resonator has mirrors of equal radius of curvature \( R \) separated by \( L = R \). The foci of both mirrors coincide at the cavity midplane.

For the confocal cavity, \( g_1 = g_2 = 0 \), so \( g_1 g_2 = 0 \) — on the boundary of stability. Despite marginal stability, it is practically useful because small misalignments do not dramatically increase loss.

\[ w_m = \left( \frac{\lambda L}{2\pi} \right)^{1/2} \left( 2g_1 g_2 - g_1 - g_2 + \ldots \right)^{1/2} \]\[ w_0 = \sqrt{\frac{\lambda L}{2\pi}} \]

6.2 Etalon as a Wavelength Selector

\[ \nu_m = \frac{mc}{2nd\cos\theta}, \quad m = 0, 1, 2, \ldots \]

By tilting the etalon or choosing \( d \) such that only one etalon peak falls within one cavity mode spacing, single longitudinal mode operation is achieved.

\[ \mathcal{F} = \frac{\pi \sqrt{R_e}}{1 - R_e} \]

determines the ratio of free spectral range to linewidth, and hence the selectivity.

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Chapter 7: Gaussian Beams

7.1 The Paraxial Wave Equation

\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + 2ik \frac{\partial u}{\partial z} = 0 \]

The simplest solution is the Gaussian beam (TEM₀₀ mode).

7.2 The Gaussian Beam

\[ E(r, z) = E_0 \frac{w_0}{w(z)} \exp\!\left( -\frac{r^2}{w(z)^2} \right) \exp\!\left( -ikz - ik\frac{r^2}{2R(z)} + i\zeta(z) \right) \]

where:

• Beam radius at position \( z \): \( w(z) = w_0 \sqrt{1 + (z/z_R)^2} \)
• Radius of curvature of the wavefront: \( R(z) = z\left[1 + (z_R/z)^2\right] \)
• Gouy phase: \( \zeta(z) = \arctan(z/z_R) \)

The intensity profile: \( I(r,z) \propto \exp\!\left(-2r^2/w(z)^2\right) \)

Key properties:

• At \( z = 0 \): minimum beam radius \( w_0 \), planar wavefront (\( R \to \infty \)).
• For \( z \gg z_R \): beam diverges as \( w(z) \approx w_0 z/z_R = \frac{\lambda z}{\pi w_0} \), half-angle \( \theta \approx \lambda/(\pi w_0) \).
• The depth of focus (confocal parameter) is \( 2z_R = 2\pi w_0^2/\lambda \).

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Chapter 8: Gaussian Beam Propagation and ABCD Matrices

8.1 The Complex Beam Parameter

\[ \frac{1}{q(z)} = \frac{1}{R(z)} - i\frac{\lambda}{\pi w(z)^2} \]

At the waist: \( 1/q_0 = -i\lambda/(\pi w_0^2) = -i/z_R \), so \( q_0 = iz_R \).

Free-space propagation by distance \( d \): \( q(z+d) = q(z) + d \).

8.2 Ray Transfer (ABCD) Matrices

\[ \begin{pmatrix} r_2 \\ r_2' \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} r_1 \\ r_1' \end{pmatrix} \]

Common matrices:

Element	ABCD matrix
Free space (length \( d \))	\( \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} \)
Thin lens (focal length \( f \))	\( \begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix} \)
Curved mirror (radius \( R \))	\( \begin{pmatrix} 1 & 0 \\ -2/R & 1 \end{pmatrix} \)
Flat interface (indices \( n_1, n_2 \))	\( \begin{pmatrix} 1 & 0 \\ 0 & n_1/n_2 \end{pmatrix} \)

8.3 ABCD Law for Gaussian Beams

\[ q_2 = \frac{Aq_1 + B}{Cq_1 + D} \]

This bilinear (Möbius) transformation fully describes Gaussian beam propagation through any sequence of paraxial optical elements.

Application — focusing a Gaussian beam:

\[ w_0' = \frac{f w_0}{\sqrt{(d_1 - f)^2 + z_R^2}} \]\[ d_2 = f + \frac{(d_1 - f)f^2}{(d_1 - f)^2 + z_R^2} \]\[ w_0' \approx \frac{f\lambda}{\pi w_1} \]

where \( w_1 = w(d_1) \) is the beam radius at the lens. To achieve the tightest focus, use a large input beam.

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Chapter 9: Lineshapes and Spectral Broadening

9.1 Homogeneous Broadening

\[ \Delta\nu_N = \frac{1}{2\pi\tau} \]\[ \Delta\nu_{pressure} = \frac{1}{\pi T_2} \]

At high pressures, collisions dominate over natural linewidth.

\[ g(\nu) = \frac{\Delta\nu/(2\pi)}{(\nu - \nu_0)^2 + (\Delta\nu/2)^2} \]

9.2 Inhomogeneous Broadening

\[ \nu' = \nu_0\left(1 + v_z/c\right) \]\[ g_D(\nu) = \frac{2}{\Delta\nu_D}\sqrt{\frac{\ln 2}{\pi}} \exp\!\left[-4\ln 2\left(\frac{\nu - \nu_0}{\Delta\nu_D}\right)^2\right] \]\[ \Delta\nu_D = \frac{2\nu_0}{c}\sqrt{\frac{2k_BT\ln 2}{M}} \]

The combination of homogeneous (Lorentzian) and inhomogeneous (Gaussian) broadening produces a Voigt profile (convolution of Lorentzian and Gaussian).

9.3 Spectral and Spatial Hole Burning

Spectral hole burning (inhomogeneous media): A strong monochromatic field saturates only those atoms whose Doppler-shifted frequency is resonant with the field. In the gain curve, a “hole” is burned at the oscillating frequency. A second weak probe experiences reduced gain only near that frequency. Allows multiple longitudinal modes to oscillate simultaneously in inhomogeneous media.

Lamb dip: In a bidirectional cavity, two counter-propagating modes interact with different velocity classes — except near line centre where both modes interact with the same \( v_z \approx 0 \) atoms. This creates a dip in the gain curve (Lamb dip) at line centre, useful for frequency stabilization.

Spatial hole burning: In a standing-wave cavity, the field intensity has nodes and antinodes along the cavity axis. The inversion is depleted at the antinodes but remains at the nodes → spatial periodicity in the population inversion → mode coupling, instabilities in single-mode operation. Eliminated in travelling-wave (ring) cavities or by using very short gain media.

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Chapter 10: Transient Behaviour and Q-Switching

10.1 Transient Laser Dynamics

\[ \frac{d\phi}{dt} = \left(\sigma c \Delta N - \frac{1}{\tau_c}\right)\phi + \frac{N_2}{\tau_{sp}} \]\[ \frac{d\Delta N}{dt} = R_p - \frac{\Delta N}{\tau_{21}} - 2\sigma c \phi \Delta N \]

where \( \tau_c = L_{eff}/(\delta_{total} c) \) is the photon lifetime in the cavity.

\[ \Omega_R \approx \sqrt{\frac{\phi_{ss}}{\tau_c \tau_{21}}} \]

Relaxation oscillation frequency is typically 100 kHz–1 MHz for solid-state lasers.

10.2 Q-Switching

\( Q = 2\pi \nu \times (\text{stored energy})/(\text{power loss}) = \omega \tau_c \).

Operating principle:

1. Low-Q phase: The cavity Q is degraded (high loss) so lasing cannot occur. The pump continues to build up the population inversion far above threshold.
2. Q-switch: The loss is suddenly removed (Q switched to high). The large stored inversion leads to rapid build-up of photon density and a giant pulse (pulse width ~1–100 ns, peak power MW–GW).

Q-switching methods:

• Rotating mirror (mechanical): simple but limited repetition rate (~100 Hz).
• Electro-optic (Pockels cell): voltage-controlled birefringence; fast (ns), precise timing.
• Acousto-optic modulator: RF-driven diffraction grating; moderate speed.
• Saturable absorber (passive Q-switch): absorber bleaches when intensity reaches threshold; self-triggered.

\[ E_{pulse} \approx h\nu \int \Delta N \, dV = h\nu (N_{2,\text{initial}} - N_{2,\text{final}}) V \]

Peak power is much higher than CW power because the energy of many round-trip times is released in a single pass through threshold.

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Chapter 11: Mode-Locking and Nonlinear Optics

11.1 Mode-Locking

\[ E(t) = E_0 \sum_{q=0}^{N-1} e^{i2\pi(\nu_0 + q\,\Delta\nu_{FSR})t} = E_0 e^{i2\pi\nu_0 t} \frac{\sin(N\pi\Delta\nu_{FSR} t)}{\sin(\pi\Delta\nu_{FSR} t)} \]

This is a pulse train with:

• Repetition rate: \( f_{rep} = \Delta\nu_{FSR} = c/(2L) \) (one pulse per round trip).
• Pulse duration: \( \Delta t \approx 1/(N\,\Delta\nu_{FSR}) = 1/\Delta\nu_{gain} \) (inverse of total gain bandwidth).
• Peak power: \( N^2 \times \) the average power.

Mode-locking methods:

• Active mode-locking: an intracavity modulator at the round-trip frequency \( f_{rep} \) periodically modulates the gain or loss → sidebands couple modes.
• Passive mode-locking (saturable absorber or Kerr lens): a nonlinear element preferentially amplifies the most intense part of the circulating field → self-consistent pulse formation.

Typical pulse durations:

Laser medium	Bandwidth	Minimum pulse
Nd:YAG	~120 GHz	~10 ps
Ti:Sapphire	~100 THz	~5 fs
Dye laser	~10 THz	~100 fs

Chirped Pulse Amplification (CPA) — Strickland & Mourou (1985, Nobel Prize 2018): To amplify ultrashort pulses to very high energy without destroying the gain medium by nonlinear effects:

1. Stretch the pulse in time by a factor \( \sim 10^3 \)–\( 10^4 \) using a diffraction grating pair (introducing frequency-dependent group delay → chirp).
2. Amplify the stretched, low-peak-power pulse in the gain medium.
3. Recompress the amplified pulse with a matched grating pair → restore short pulse with greatly increased energy.

This enables table-top lasers with peak intensities \( > 10^{20} \, \text{W/cm}^2 \).

11.2 Introduction to Nonlinear Optics

11.2.1 Why Lasers Enabled Nonlinear Optics

\[ P = \varepsilon_0 \left( \chi^{(1)} E + \chi^{(2)} E^2 + \chi^{(3)} E^3 + \ldots \right) \]

The ratio of nonlinear to linear response is \( \chi^{(2)} E / \chi^{(1)} \sim E/E_{at} \), where \( E_{at} \sim 10^{11} \, \text{V/m} \) is the atomic field. Thermal light sources have \( E \ll E_{at} \); laser beams can approach and exceed this.

First observation: Franken et al. (1961) frequency-doubled a ruby laser in quartz crystal — detected UV at twice the optical frequency. This confirmed \( \chi^{(2)} \neq 0 \) for non-centrosymmetric crystals.

11.2.2 Second-Harmonic Generation (SHG / Frequency Doubling)

\[ P^{(2)}(2\omega) = \varepsilon_0 \chi^{(2)} E(\omega)^2 \]

This acts as a source for a new field at \( 2\omega \).

\[ \Delta k = k(2\omega) - 2k(\omega) = 0 \]

Since most materials are normally dispersive (\( n(2\omega) > n(\omega) \)), phase matching requires birefringent phase matching (using different polarization directions in anisotropic crystals) or quasi-phase matching (periodic poling of the nonlinear coefficient).

Conversion efficiency grows as \( L^2 \) (for perfect phase matching) where \( L \) is the crystal length, making high intensity (tight focusing) and long crystals advantageous.

11.2.3 Other \( \chi^{(2)} \) and \( \chi^{(3)} \) Processes

Process	Order	Description
SHG	\( \chi^{(2)} \)	\( \omega + \omega \to 2\omega \)
Sum-frequency generation (SFG)	\( \chi^{(2)} \)	\( \omega_1 + \omega_2 \to \omega_3 \)
Difference-frequency generation (DFG)	\( \chi^{(2)} \)	\( \omega_1 - \omega_2 \to \omega_3 \)
Optical parametric amplification (OPA)	\( \chi^{(2)} \)	Amplifies \( \omega_s, \omega_i \) from pump \( \omega_p \)
Self-phase modulation (SPM)	\( \chi^{(3)} \)	Intensity-dependent phase → spectral broadening
Stimulated Raman scattering	\( \chi^{(3)} \)	Inelastic photon scattering via molecular vibrations
Kerr effect (intensity-dependent \( n \))	\( \chi^{(3)} \)	\( n = n_0 + n_2 I \) → self-focusing, filamentation

The Kerr effect in optical fibers (\( n_2 \approx 2.6 \times 10^{-20} \, \text{m}^2/\text{W} \)) combined with dispersion management is the basis of soliton propagation and ultrafast fiber lasers.